In this talk, I will introduce some generalization of Pitman’s transform and show that the dynamics of several discrete integrable systems, such as the discrete Korteweg-de Vries equation, the ultra-discrete Korteweg-de Vries equation, the ultra-discrete Toda equation, the box-ball system, are given by them. We apply this observation to define infinite space versions of these models and study their invariant measures using the generalized Pitman’s transform. This talk is based on a joint work with David Croydon, Tsuyoshi Kato and Satoshi Tsujimoto.